Active transducer



April 9, 1957 J. G. LlNVlLL 2,788,496

ACTIVE TRANSDUCER Filed June 8, 1953 2 Sheets-Sheet 1 F IG.

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/N VENTOR NEGATIVE J. G. L/NV/LL IMPEDANCE CONVERTER RL BY k WfWATTORNEY April 9, 1957 J. G. LlNVlLL 2,788,496

ACTIVE TRANSDUCER Filed June 8, 1953 2 Sheets-Sheet 2 FIG/3 51 W 2 /7 IR6 5 AvAvAv 5 u u NEGATIVE s IMPEDANCE 5 CONVERTER C? L.

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l 3/ 30 3g a4 NEGATIVE NEGATIVE lMPDANCE AMP 'IMPEDANCE I 5; CONVERTERCONVERTER L J 1 0 Y J a 6 is 27 26 lNl ENTOR J G. LIN V/LL ATTORNEYnited States Patent "ice 2,788,496 it ACTIVE TRANSDUCER John G. Linvill,Whippany, N; 1., assignor to Bell Telephone Laboratories, Incorporated,New York, N. Y., a corporation of New York Application June 8, 1953,Serial No. 360,301

3 9 Claims. (c1. ass-so procedure for an active transducer having anunrestricted transmission characteristic. Other objects are to reducethe loss, or even provide a gain, in the transmission band of an activewave filter. Another object is to eliminate one type of reactor in atransducer without restrictingits transmission characteristic. Anotherobject is to reduce the number of component reactors required. Furtherobjects are to decrease the size and cost of a transducer which mustmeet particularly severe transmission requirements.

The network designer oftenencounters transmission requirements which hecannot meet economically in a passive transducer. For example, a wavefilter which will transmit very low frequencies usually requires verylarge and expensive inductors. For this case, a passive transducer madeup of only resistors and capacitors,

usually called anR-C network, .is attractive. However, for a givencharacteristic, an R-C filter introducesex cessive loss in thetransmission band and requires many more elements than does a filterwhich includes both inductors and capacitors. These defects may beovercome by including an active element. One active transducer of thistype is disclosed in United States Patent 2,549,065, issued April 17,1951, to R. L. Dietzold. .The active element is a stabilized feedbackamplifier. The

passive elements are resistors and only one type of reactor, eithercapacitive or inductive.

The present invention is directed to another type of active transducerwith unrestricted transmission characteristic. The circuit comprises twopassive networks and a negative impedance converter connected in tandembetween them. The negative impedance converter, hereafter called simplya converter, has an impedance conversion ratio, designated M, which isnegative. Thus, it presents at one pair of its terminals an impedancewhich is M times the impedance connected to its other terminal pair.Each of the passivenetw orks is made up of one or more resistors and oneor more reactors. The reactors may include both inductors andcapacitors, or they may be all of the same type. Usually, it ispreferred to use only capacitors for the reactors. Resistors andcapacitors are, in general, cheaper, smaller, and more nearly idealelements than are inductors. The passive networks may be simplestructures, either balanced or unbalanced. They may, for example, belattice, ladder, bridged-T, or twin-T networks. 7 reactors required inthe networks is no greater than that required in apassive transducerhaving a comparable transmission characteristic. 3

At their ends facing the converter, the passive networks havedriving-point impedances oneof which is equal to -M times the other atoneor more preselected active transducer in accordance with theinvention.

2,788,496 Patented Apr. 9, 1957 natural frequencies of the transducer.Thenatural frequencies are the complex frequencies at which the transferimpedance of the transducer is infinite. The transfer impedance of thetransducer is unrestricted as to roots and poles in that-any physicallyrealizable transmission characteristics may be provided. This is trueeven when the passive networks are restricted to reactors of only onetype. As examples, lowpass, high-pass, and bandpass wave filtersaredisclosed. The loss in the trans mission band may be reduced by theactive portion of the transducer. Also, a gain in the band may beprovided by proper design of t he converter or by inserting an amplifierin tandem between two active transducers.

The nature of the invention and its various objects, features, andadvantages will appear more fully in the following detailed descriptionof preferred embodiments illustrated in the accompanying drawing, ofwhich Fig. l is a block diagram of an active transducer in accordancewith the invention;

Fig. 2 represents the complex frequency plane it on which a re plottedthe poles of the transfer impedance of a typical low-pass filter inaccordance with the invention;-

Fig. 3 shows, on the complex frequency plane, the distribution of thezeroes and the poles of the difierence between the driving-pointimpedances of the passive networks at their ends facing the converter,for the low-pass filter example; e n

Fig 4 shows the circuitof the low-pass filter when the passive networksare unbalanced ladder structures made up of resistors and shuntcapacitors; r i

Fig. 5 is atypical relativeresponseversus frequency characteristicobtainable with the low-pass filter of'Fig. 4; Fig. 6 shows the circuitof a high-pass filter in accordance with the invention, in which thepassive networks are constituted by resistors and shunt inductors;

Figs. 7, 8, and 9 show, respectively, the pole-zero dis tribution, thenetwork configuration, and a typical characteristic of a secondhigh-pass filter in accordance with the invention;

Fig. 10 shows two active band-pass filters in accordance with theinvention connected in tandem by an amplifier;

Fig. 11 shows the relative response characteristic of the filter of Fig.10;

Fig. .12 shows a generalized lattice structure which may be used for thepassive networks in Fig. 1; and

Figs. 13 and 14 show, respectively, the network configuration and thecharacteristic of a low-pass filter in accordance with the invention inwhich a twin-T network provides a peak of attenuation at a finitefrequency.

Fig. 1 shows in block diagram an embodiment of an The transducercomprises two four-terminal passive networks 1 and 2 and an interposednegative impedance converter 3 connected in tandem between a pair ofinput terminals 5, 6. and a pair of output terminals 7, 8. A suitablesource of signals, not shown, may be connected to the The total numberof input terminals and a suitable load, not shown, may be connected tothe output terminals. l

The converter 3 is an active four-pole network which presents at itsinput terminals 9, 10 an i impedance which, over the frequency range ofinterest, is equal to M times the impedance connected to its outputterminals 11, 12. The converter 3 may be of thevacuum-tube type,examples of which are disclosed in the paper by J. L. Merrill, Jr.,entitled, Theory of the negative impedance converter, in the Bell SystemTechnical Journal, vol. XXX, No. 1, January 1951, pages 88 to 10 9.Preferably, however, the converter 3 is of the transistor type, suitablecircuits of which are disclosed in the copending United States patentapplication of R. L. Wallace, Jr.,

and the present applicant, Serial No. 310,084, filed September 17, 1952,now Patent No. 2,726,370 issued December 6, 1955. A converter using oneor more transistors is preferred because it may be designed to have morenearly ideal characteristics. Therefore, a transducer employing aconverter of this type may be designed to meet a prescribed transmissioncharacteristic within closer limits, and the characteristic will be morestable with time.

The converter 3 has a current transfer ratio Mi, which is the ratio ofthe input current'lato the output In, and a voltage transfer ratio Me,given by the ratio of the input voltage Ea to the output voltage Eb. Oneof these ratios is always negative. The ratio M1 is substantially unityif junction transistors are used in the converter. Its value may beapproximately doubled by using point contact transistors, but at asacrifice in the stability of the converter. This ratio may, of course,be extended by associating a transformer with the converter. Asexplained in the above-mentioned patent application, the magnitude of Memay be selected within wide limits.

The impedance conversion ratio M of the converter, which is the ratio ofMe to Mt, may be given any negative value within a Wide range bychoosing appropriate values of Me and Mi. A judicious choice of M mayhelp in obtaining convenient values for the component impedance elementsin the passive networks 1 and 2.

It is obvious that a given impedance conversion ratio M may be obtainedwith an infinite number of combinations of the ratios Ml and Me.However, their choice determines the gain factor G, which is the productof M1 and Me. If the networks 1 and 2 and the ratio M all remain fixed,the power gain of the transducer from input to output is inverselyproportional to the magnitude of G.

The theory of the design of a transducer of the type shown in Fig. 1using such a converter will now be presented. The transfer functions ofa four-terminal transducer made up of lumped impedance elements arerotational functions of the frequency 7". The analysis is simplifiedbythe introduction of the parameter p, called the complex frequency anddefined as where ois the real part, in is the imaginary part, and w isthe radian frequency 2117. A fuller discussion of the concept of complexfrequency and the complex frequency plane may be found, for example, inchapter II of the book Network Analysis and Feedback Amplifier Design,by H. W. Bode, published by D. Van Nostrand Company, New York, 1945 Thetransfer impedance Z'rof any four-terminal network may be expressed asthe ratio of two polynomials in P giving The transfer impedance becomesinfinite at the complex frequencies at which the denominator D(p) iszero. Therefore, these complex frequencies are the natural frequenciesof the network. In passive R-C network-s, or R-L networks (comprisingonly resistors and inductors), these zeroes are restricted to thenegative real axis of the complex frequency plane. This constraintseriously limits the quality of approximation to an ideal filtercharacteristic obtainable if N(p) and D(p) are polynomials of a limiteddegree. Active R-C or R-L networks, however, can have naturalfrequencies anywhere in the left-half plane, the same as passivenetworks comprising resistors, inductors, and capacitors.

In the ideal case, if the current transfer ratio Mt is unity, thecurrent Ia entering the converter 3 is equal to the current Ib leavingit, that is,

lo=lb 3 Assuming a voltage transfer ratio Me of l, the input voltage Eais the negative of the output voltage, that is,

output terminals 7, 8, which is the ratio of the output voltage E2 tothe input current I1 when the output current I2 is zero, is given by InEquation 5, Z122. is the transfer impedance of the network 1, Z121; isthe transfer impedance of the network 2, 2225 is the driving-pointimpedance of the network 1 at the terminals 9, 10, and Zllb is thedriving-point impedance of the network 2- at the terminals 11, 12. Thederivation of Equation 5 involves, as an intermediate step, theevaluation'of the input current In. to the converter 3. The load at theterminals 9, 10 of the network 1 is the input impedance Z: seen at theterminals 9, 10 of the converter 3, given by ZI=MZ11b where M is theimpedance conversion ratio of the converter. When M is -1, as assumedhere,

ZI=-Z11b Therefore, in accordance with network theory,

I Z I 1 12a 8 a 22Q 11b But since the current Ib flowing into thenetwork 2 is equal to la, the output voltage E2 is It will be observedfrom Equation 11 that, in the present case whereMe and Mi are each ofunit magnitude, the transducer obeys the law of reciprocity only inmagnitude. The transfer impedance changes sign when the input and outputterminal pairs are interchanged.

The driving-point impedance Z11 of the transducer at the input end is EZ 0 2 Z Z 11 I1 no 22a llb where E1 is the input voltage and Zlla. isthe driving-point impedanceof the network 1 at the terminals 5, 6. Thedriving-point impedance Z22 at the output end of the transducer is whereZ221; is the driving-point impedance of the network 2 at the terminals7, 8.

In Equation 2, the zeroes of the numerator N(p) are associated with thestructure of the network, not with its natural frequencies. Forinstance, ladder networks have zeroes of transfer impedance atfrequencies where shunt elements become short circuits or serieselements become open circuits, Thus, an R-C ladder-type network. made upof resistors and. three. shunt; capacitors will have three zeroes oftransfer impedance at infinite frequency, where the capacitors are shortcircuits, irre spective of the values of the capacitors or the naturalfrequencies of the network. Lattice, bridged-T, and twin-T networks willhave zeroes of transmission at frequencies where a bridge-like balanceoccurs, regardless of the location of the natural frequencies of thecomplete network.

A suggested procedure for designing an active transducer in accordancewith the invention comprises three steps. First, the designer prescribesthe desired transfer impedance, within a constant multiplier, in theform of Equation 2. Next, he selects the zeroes of the denominator D(p)as the natural frequencies of an active network of the type shown inFig. 1. Then, by a method to be described below, he obtains for thenetworks 1 and 2 a pair of driving-point impedances Z2221, and Zllbwhich are consistent with these natural frequencies. Finally, hesynthesizes networks which have the driving-point impedances Z223. andZllb. The networks chosen must also be of a form which will provide thedesired zeroes of transmission at the zeroes of the numerator N(p).

To illustrate the application of the procedure outlined above, thedesign of several active wave filters of the form shown in Fig. 1 willnow be described.- The first example is a low-pass filter which has aButterworth characteristic, a cut-off frequency-1 fc of 1000 cycles persecond, and an attenuation rising at the rate of 18 decibels per octaveof frequency. In the form of Equation 2, the transfer impedance of suchafilter. may be written as N (P) K 1m) r (if 21 21r +2 21 21rf where Kis a numerical constant which determines the impedance level of thefilter. The transfer impedance will have three poles, corresponding tothe zeroes of D(p), and three zeroes. The. zeroes all fall at infinitefrequency. The poles are known to fall on a semicircle in the left halfof the complex frequency plane. Fig. 2

shows the poles Pa, $5., and Pb plotted on a complex frequency plane inwhich the real (or tr) axis is horizontal and the imaginary (or its)axis is vertical. The coordinates of these poles are Pa=21r500+j21r866Fa=-21r500j21r866 (16) Pb= 21r1000+j0 (17) It is seen that the poles Paand P: are conjugates and the pole Pb falls on the negative real axis.

The natural frequencies of the filter must occur at these complexfrequencies. The circuit shown in Fig. 1 will have natural frequencieswhen the driving-point impedance Z2211 of the network 1 and thedriving-point impedance Zllb of the network 2 are equal. At each ofthese frequencies, the sum Z of the impedance -Z1lb seen looking intothe converter 3 at its input terminals 9, 10 and the impedance Z222. iszero, that is, when The zeroes of Z will, therefore, be at the complexfretmf where D(p) isithe denominator in Equation 14. selection ofthe'points 0'1, 02, and as is arbitrary as far as the transfer impedanceZ21 of; the filter is concerned. Of course, none should coincide withthe pole Pb. However, as explained below, the driving point impedancesand othercharacteristics of the filter are influenced by the selection.

There remains, now, only the synthesis of the passive R-C networks 1 and2. A suggested method involves, first, the expansion of Equation 19 inpartial fractions. It is found that the residues are always real but maybe positive or negative. The terms are divided into a first group withpositive residues and a second group with negative residues. It is knownthat any function with simple poles on the negative real axis andpositive real residues in those poles is the driving-point impedance ofan R-C network. Therefore, the first group of terms is associated withthe impedance Zen. of the network 1. The second group of terms isassociated with the impedance Zllb of the network 2. Although thenetwork 2 can only provide positive residues in the poles of Ziib, theywill appear as negative residues when viewed from the input terminals 9,10 of the converter 3. It is now assumed that the networks 1 and 2 willbe ladder-type structures comprising resistors and shunt capacitors. Therequired values of the component elements are found by making a Cauersynthesis. I For the distributions of the critical frequencies Pa, Pa,Pb, 0,, (r and 0' shown in Fig. 3, the networks 1 and 2 will have theconfigurations shown in Fig. 4. The network 1 comprises three seriesresistors R1, R2, and R3 and. two shunt capacitors C and C2. The network2 consists of the parallel combination of a resistor R4 and a capacitorC3. The fact that the forms shown for the networks 1 and 2 are correctcan be understood from the following analysis. From the distribution ofthe poles of Z shown in Fig. 3, it can be observed that the residues inthe poles at a, and a, are positive while the residue in the pole at ais negative. Therefore, the constant K, appearing in Equation 14, andthe partial fractions associated with the poles at a, and a, areidentified with the impedance Z222. of the network 1 and the partialfraction associated with the pole a, is identified with the impedanceZnbas seen through the converter 3. Accordingly, the network 1 will includetwo capacitors and the network 2 only one capacitor. The required valuesof the resistors R1, R2, R3, and R4, in ohms, are .600, 2040, 1200, and1300, respectively, and the values of the capacitors C1, C2, and C3, inmicrofarads, are 0.384, 0.172, and 0.0820, respectively. It is assumedthat the impedance conversion ratio M of the converter 3 is l. Fig. 5shows a typical relative response characteristic obtainable with thelow-pass filter of Fig. 4. It is assumed that the signal source ofvoltage E1 connected to the terminals 5, 6 has zero internal impedance.if the source has an equivalent series impedance R0, the series resistorR1 is replaced by a resistor of smaller value R1 given by R1'=R1R0 (20)In Fig. 5, the ratio of the output voltage E2 to the input voltage Ei,expressed in decibels, is plotted against the frequency in cycles persecond, on a logarithmic frequency scale. The. response relative to thatat zero frequency is shown. By properly choosing the gain factor G ofthe converter 3, there may be provided any gainwhich is consistent withthe stability'of the con verter and the networks 1 and 2.

i Fig. 6 shows the circuit of a second wavefilter'in accordance with theinvention. This is a high-pass filter having a Butterworthcharacteristic and an attenuation which rises at the rate of 18 decibelsper octave. The passive networks 1 and 2 are ladder-type RL structures,comprising resistors and shunt inductors. Their configurations are thesame as those shown for the low-pass filter of Fig. 4 except that thethree shunt capacitors C1, C2, and C3 are replaced, respectively, by thethree shunt inductors L1, L2, and L3. Otherwise, the circuit of Fig. 6is similar to that of Fig. 4. The component resistors and inductorsrequired in the filter of Fig. 6 may be evaluated by the same proceduredescribed above in connection with the filter of Fig. 4.

Figs. 7, S, and 9 relate to another high-pass wave filter in accordancewith the invention. The filter has a Butterworth characteristic with anattenuation rising at the rate of 24 decibels .p'er octave. On thecomplex frequency plane of Fig. 7, the points 14, 15, 16, and 17 markedby xs are poles of the transfer impedance Z21 and zeroes of theimpedance Z. These points all fall on a semicircle 23 in the left halfof the plane and have a uniform spacing S from the origin. They alsohave a uniform angular spacing U of 45 degrees on the semicircle 23, andthe points 14 and 17 have equal angular spacings of U/2 from the foraxis. Thus, the points 14 and 17 are conjugate, as are also the pointsand 16.

The points 19, 20, 21, and 22 marked by +s are poles of Z. These polesall fall on the negative real axis and have spacings of S1, S2, and S3,as shown. The displacement of the nearest pole 22 from the origin is St.The impedance Z21 has a fourth-order zero at the origin, as shown by thecircle 24-.

Fig. 8 is a schematic circuit of a filter having the distribution ofpoles and zeroes shown in Fig. 7. Each of the passive networks 1 and 2is an RC ladder structure comprising two series capacitors and threeshunt resistors. The procedure described in connection with Fig. 4 maybe used to find the required values of these elements. Fig. 9 shows arelative response characteristic obtainable with the filter of Fig. 8when the source E1 and the load Rn each have a high impedance comparedto the resistance of the end resistor connected in parallel therewith.

In accordance with the invention, two or more active transducers may beconnected in tandem and isolated from eachother by one or moreamplifiers. As an example, Fig. 10 shows two band-pass wave filters 25and 26 connected in tandem between input terminals 5, 6 and outputterminals 7, 8 and an interposed amplifier 27, preferably of thetransistor type. The filter 25 comprises two passive networks 29 and 30connected through a converter 31. In the filter 26, the correspondingunits are designated 33, 34, and 35. Each of the networks 29 and 33 isconstituted by the series combination of a resistor and a capacitor in aseries branch. Each of the networks 30 and 34 is made up of the parallelcombination of a resistor and a capacitor in a shunt branch.

Fig. 11 shows a typical relative response characteristic obtainable withthe filter of Fig. 10. The midband frequency is located at 1000 cyclesper second and the band width is approximately 200 cycles. Zeroes oftnan-smission occur at zero and infinite frequencies. By properlydesigning and adjusting the amplifier 27, considerable gain in thetransmission band may be achieved, if desired.

In the filter circuits shown in Figs. 4, 6, 8, and 10, the zeroes oftransmission occur only at zero or infinite frequency. With the RC andL-C ladder-type passive networks used in these filters, it is impossibleto obtain a zero of transmission at a real frequency between zero andinfinity. However, in accordance with the inven- 8 tion,-intermediatezeroes of transmission (peaks of attenuation) may be provided by usingR-C or L-C lattice networks, or unbalanced equivalents thereof such asbridged-T or twin-T structures.

Fig. 12 shows the generalized circuit of a lattice structure which maybe used for either or both of the passive networks 1 and 2 in Fig. 1.The lattice comprises two equal series impedances Za, Za and two equaldiagonal impedances Zb, Zb connected between a pair of input terminals37, 33 and a pair of output terminals 39, 40. The synthesis of an activefilter of the type shown in Fig. 1 using two such lattice networks toprovide intermediate attenuation peaks proceeds as explained above inconnection with Fig. 4 through the specification of the desired transferimpedanceZzi, the selection of the impedance Z, and the evaluation ofthe driving-point im pedances Z229. and Z1111. At this point, one hasthe driving-point impedances of the networks 1 and 2 and knows thefrequencies at which these networks should introduce peak-s ofattenuation.

The driving-point impedance Zn at either end of the lattice of Fig. 12is D 2 B p) and the transfer impedance Zr in either direction is Z I) Za T p Z 23 It is already known that the driving-point impedance of thenetwork 1 will be Z223. and the driving-point impedance of the network 2will be 2111;. When one substitutes expressions for the driving-pointand the transfer impedances for the lattices, as given in Equations 22and 23, into an equation of the form of 5, the numerators of theexpressions for Z122. and ZlZb are found to include constantmultiplier-s and all of the factors of N(p) but no others. This is sobecause the drivingpoint and the transfer impedances of a lattice havethe same denominator B(p). These factors of N(p) are divided inconjugate pairs between the numerators of Z12a and Z121) in any waywhich makes each numerator of no higher degree than the associateddenominator. One thus obtains an expression for the transfer impedanceof each of the lattices 1 and 2 in the form of Equation 23 in which thefactors of the numerator T(p) are specified but a constant multiplier isyet to be determined. Expressions for the impedances Za and Zb of eachof the lattices 1 and 2 may now be found by adding Zn to Zr to obtain Zband subtracting Zr from Zn to get Za. Now, if the i-mpedances Za and Zbare to be R-C structures, one selects the largest permissible values forthe constant multipliers. Finally, he determines the configurations ofthese branches and the required values of the component resistors andcapacitors to provide the impedances Za and Zb. The synthesis of an RClattice network from a specified drivingpoint impedance and a transferimpedance specified within a constant multiplier is described in greaterdetail, for example, in the paper by J. L. Bower and P. F. Ordungentitled, The synthesis of resistor-capacitor networks, in theProceedings of the I. R. B, vol. 38, No. 3, March 1950, pages 263 to269.

It is often desirable to convert the lattices thus found for thenetworks 1 and 2 into equivalent unbalanced structures. However, it isusually difiicult, if not impossible, to predict the configuration ofthe equivalent unbalanced network from the form of the lattice. When itis desired to obtain, for one of the networks 1 and 2, a particular typeof unbalanced structure, one may use a modified method which avoidsfirst finding a lattice and then seeking the unbalanced equivalent. Thismethod, which is applicable to either R-C or -RL networks, involvesexchanging part of the inherent latitude inthe choice of poles of theimpedance Z for a constraint in the form of one of the networks 1 and 2.

19 There will now be presented, as an example of this modified method,an active low-pass wave filterg-cf the type shown in Fig. 1 in which anunbalanced, twin-T, R-C network is used to provide a peak of attenuationat a finite frequency. As shown in Fig. 13, the passive network 1comprises a resistor R5 in series with a parallel twin-T structure. Oneof the Ts is constituted by the two equal series resistors R6, R6 and aninterposed shunt capacitor C4. The other is made up of the equal seriescapacitors C5, C5 and the interposed shunt resistor R7. The network 2 isan R-C ladder structure with a series resistor, a shunt resistor, andtwo shunt capacitors which will provide a 12-decibel per octaveattenuation rate at the higher frequencies. Fig. 14 gives the relativeresponse characteristic of the filter. The band cuts off at 1000 cyclesper secohd. The attenuation peak occurs at 2000 cycles.

As in the previous examples, the first step in designing the circuit ofFig. 13 is to select the poles and zeroes of the transfer impedance Z21of the filter to provide an acceptable characteristic, that is, onehaving a reasonably fiat pass band from zero to 1000cycles, anattenuation peak at 2000 cycles, and a high-frequency attenuation rateof 12 decibels per octave. The method described in my paper entitledyTheapproximation with rational functions of prescribed magnitude and phasecharacteristics, in the Proceedings of the I. R. E., vol. 40, No. 6,June 1952, pages 711 to 721, may be followed in determining thelocations of the poles and zeroes.

In the network 1, the initial resistor R5 .is selected in accordancewith the impedance of the voltage source E1. The twin-T structure isdesigned to have reasonable values for the component elements Re, R6,R7, C4, C5, and C5 and to provide the attenuation peak at 2000 cycles. Amethod of designing such a twin-T network is presented, for example, inUnited States Patent 2,106,785, to H. W. Augustadt, issued February 1,'1938.

Next, the driving-point impedance Z22. of the net- Work 1 at its endfacing the converter 3 is expressed in the form of a partial fractionexpansion. Then, the driving-point impedance Z1111 of the network 2 atits end facing the converter 3 is determined. This can be done through aconsideration of the expression for Z, which, as given by Equation 18,is the difference between Z22a and Z111). One knows certain factors of Zat this point. Its numerator has a constant multiplier K1 and includesthe factors of D(p). Its denominator includes as factors the denominatorof Z229. and the denominator of Zub, as yet undetermined but known to beof the form (p-l-a) (p+b), Where a and b are as yet unknown. One writesan expression for Z, employing the known factors and putting the unknownfactors in literal form, and. expands this expression in partialfractions. By known mathematical methods, the unknown constants K1, a,and b are selected consistent with the requirements that the residues inthe poles of Z2211 must be positive and of the values already known andthat the residues in the remaining poles, associated with Z11b, must benegative. With the constants K1, a, and b determined, one may write anexplicit expression for Z. The impedance Zllb is found by substracting Zfrom Z22a- With Ziib thus determined, the final step is to synthesize anR-C ladder structure with shunt capacitors to obtain the network 2 shownin Fig. 13.

The synthesis procedure described above in connection with Fig. 4results in a filter having a desired prescribed transmissioncharacteristic. However, it does not lead to unique structures for thepassive networks 1 and 2 because of the permissible latitude in thechoice of the poles of the impedance Z. An infinite number ofembodiments of the networks 1 and 2 may be found which will provide thedesired transmission characteristic for the filter as a whole. Thedifferent filters will, however, differ in their driving-pointimpedances and also in the effects on the .transmissioh characteristicof imperfec' tions in the converter 3. w r I In some applications, it isimportant to be able to prescribe the driving-point impedance of thefilter at one or both of the pairs of terminals 5 6 and 7-$. Part of thelatitude in the choice of the poles of Z may be used to obtain amoredesirable driving-point impedance at one or both of the ends of thefilter. Of course, any desired impedance transformation Within thetransducer may be provided by properly choosing the impedance conversionratio M of the converter 3. r

As seen in Equation 6, the input impedance Zr of the converter 3 dependsupon the impedance conversion ratio M. In a practical converter, theratio M changes with time, temperature, or other environmentalconditions, causing a corresponding change in Z1.- Equation 5 shows thata change in 'ZI from the value Z11b will not affect the zeroes of thetransfer impedance Z21 of the filter but will change the poles. A changein the locations of these poles will-cause a change in the transmissioncharacteristic of the filter. For example, moving a pole such as 14 inFig.7 toward the its axis to a new position which is a fractional part Hof its original distance from this axis will cause a maximum change I inthe response characteristic of the filter given in decibels by Amathematical analysis shows that, for a given change in the ratio M, theresu1ting'shift.in pole location and consequent change: T in the filtercharacteristic are dependent upon the distributionof the poles of theimpedance Z on the negative real complex frequency axis o'. From thestandpoint of drift of the filter characteristic, a .good practical ruleto apply is as follows: Assuming that Z has W poles, one is spaced fromthe origin by a distance Q which is between half and triple the averagedisplacement of the zeroes of Z from the origin, another is located notmore than Q/W from the origin, and the remaining poles are spacedapproximately uniformly between these two. A somewhat involvedmathematical analysis, here omitted, shows that in general the driftwill be less with such a distribution of poles than if eitherof the endpoles falls outside of the limits given, even though the spacing betweenpoles is kept uniform. The analysis further shows that a nonuniformspacing, especially one in which adjacent poles have a spacing much lessthan the average, aggravates the drift problem. In Fig. 7, it is seenthat the distribution of the four poles 19, 20, 21, and 22 of theimpedance Z falls Within the rule.. The distance Q from the origin ofthe most remote pole 19 is greater than 8/2 but not more than 38, whereS is the radius of the semicircle 23 on which all of the zeroes 14, 15,16, and 17 of Z fall. Also, the pole spacings S1, S2, and S3 are:approximately equal.

It is to be understood that the above-described arrangements areillustrative of the application of the principles of the invention.Numerous other arrangements may be devised by those skilled in the artwithout departing from the. spirit and scope of the invention.

What is claimed is:

1. An active transducer. comprising two passive networks and a negativeimpedance converter connected in tandem therebetween, said networkshaving at their ends "facing said converter driving-point impedanceswhich, at

a prescribed natural frequency of said transducer, are related to eachother by a numerical factor equal in magnitude to the impedanceconversion ratio of said converter.

2. A transducer in accordance with claim 1 having a transfer impedancewith unrestricted zeroes and poles, said networks comprisingresistorsand only a single type of reactor.

3. A transducer in accordance with claim 2 in which said single type ofreactor is capacitive.

4. A transducer in accordance with claim 2 in which said single type ofreactor is inductive.

5. A transducer in accordance with claim 1 in which said impedanceconversion ratio is approximately 1.

6. A transducer in accordance with claim 1 in which said impedanceconversion ratio has a magnitude greater than unity.

7. A transducer in accordance with claim 1 said converter is of thevacuum-tube type.

8. A transducer in accordance with claim 1 in which said converter is ofthe transistor type.

9. A transducer in accordance with claim 1 having the transmissioncharacteristic of a low-pass filter.

10. A transducer in accordance with claim 9 in whicl said transmissioncharacteristic has a peak of attenuation at a finite frequency otherthan zero.

11. A transducer in accordance with claim 1 having the transmissioncharacteristic of a band-pass filter.

12. A transducer in accordance with claim 1 having the transmissioncharacteristic of a high-pass filter.

13. A transducer in accordance with claim 1 having the transmissioncharacteristic of a wave filter with a peak of attenuation at a finitefrequency other than zero.

14. A transducer in accordance With claim 1 in which one of said passivenetworks is a twin-T structure.

15. A transducer in accordance with claim 14 in which said twin-Tstructure has a peak of attenuation at a finite frequency other thanzero.

16. A transducer in accordance with claim 14 in which said twin-Tstructure comprises only resistors and capacitors.

17. A transducer in accordance with claim 14 in which said twin-Tstructure comprises two T-networks connected in parallel, one of saidT-networks including two series resistors and an interposed shuntcapacitor and in which the other of said T-networks including two seriescapacitors and an interposed shunt resistor.

18. A transducer in accordance with claim 1 in which one of said passivenetworks is a ladder-type structure.

19. A transducer in accordance with claim 18 in which said ladder-typestructure comprises only resistors and capacitors.

20. A transducer in accordance with claim 18 in which said ladder-typestructure comprises only resistors and inductors.

21. A transducer in accordance with claim 18 in which said ladder-typestructure comprises a series resistor and a shunt capacitor.

22. A transducer in accordance with claim 18 in which said ladder-typestructure comprises a series resistor and a shunt inductor.

23. A transducer in accordance with claim 18 in which said ladder-typestructure comprises a series capacitor and a shunt resistor.

24. A transducer in accordance with claim 1 in which each of saidpassive networks is a ladder-type structure.

25. In combination, two transducers in accordance with claim 1 and anamplifier connected in tandem therebetween.

26. A transducer in accordance with claim 1 in which the differencebetween said driving-point impedances is an impedance having a pluralityof zeroes and a plurality of poles, said poles being W in number, all ofsaid poles being located on the negative real axis of the complexfrequency plane, one of said poles being spaced from theorigin by adistance Q which is between half and triple the average displacement ofsaid zeroes from said origin, a, second of said poles being located notmore than Q/ W from said origin, and the remaining poles being spacedapproximately uniformly between said first and said second poles.

, 27. A transducer in accordance with claim 26 in which said zeroes fallapproximately on a semicircle in the left half of said plane.

28. A transducer in accordance with claim 27 in which said zeroes areapproximately equally spaced on said semicircle.

29. A transducer in accordance with claim 28 in which two of said zeroesare conjugate.

36. A transducer in accordance with claim 1 in which said impedances arealso related to each other by said ratio at a second prescribed naturalfrequency of the transducer.

31. A transducer in accordance with claim 1 in which said impedances arealso related to each other by said ratio at a plurality of otherprescribed natural frequencies of the transducer.

32. A transducer comprising two passive networks and an interposednegative impedance converter connected in tandem, said converter havingan impedance conversion ratio approximately equal to -1, said networkscomprising resistors and only a single type of reactor, and saidnetworks having at their ends facing said converter driving-pointimpedances which are substantially equal at a preselected naturalfrequency of the transducer.

33. A transducer in accordance with claim 32 in which said single typeof reactor is capacitive.

34. A transducer in accordance with claim 32 in which said single typeof reactor is inductive.

35. A transducer in accordance with claim 32 in which one of saidnetworks is of the ladder type.

36. A transducer in accordance with claim 32 in which each of saidnetworks is of the ladder type.

37. A transducer in accordance with claim 32 in which said impedancesare also substantially equal at a second preselected natural frequencyof the transducer.

38. A transducer in accordance with claim 32 in which said impedancesare also substantially equal at a plurality of other preselected naturalfrequencies of the transducer.

39. A transducer in accordance with claim 32 in which the differencebetween said driving-point impedances is an impedance having a pluralityof zeroes and a plurality of poles, said poles being W in number, all ofsaid poles being located on the negative real axis of the complexfrequency plane, one of said poles being spaced from the origin by adistance Q which is between half and triple the average displacement ofsaid zeroes from said origin, a second of said poles being located notmore than Q/W from said origin, and the remaining poles being spacedapproximately uniformly between said first and said second poles.

References Cited in the tile of this patent UNITED STATES PATENTS2,093,665 Tellegen Sept. 21, 1937 2,197,348 Roberts Apr. 16, 19402,243,440 Roberts May 27, 1941 2,549,065 Dietzold Apr. 17, 1951

